Your second point, that we should work with Putin's Russia very carefully and in places where we have common National interests is absolutely correct, we don't have to be nice guys, just careful and honest.
Around the world and in different historical periods the word "conservative" has meant many different things, including "royalist", "imperialist", "national socialist", "Roman Catholic" & "fascist dictator".
So it's extraordinarily important that we not let American conservatives, who are a very different breed of political animal, be lumped by the leftist media in with those others.
American conservative means two things, basically: constitution and Bible, not necessarily in that order.
It also certainly means nationalist, but only in the sense of strong national defense & law enforcement, not with the idea of military adventurism.